Broadcast live on the BBC, the vignette concluded with the SNP leader beseeching Miliband to ally against the common enemy in Downing Street, a proposal the now-former Labour chief rebuffed. Earlier, Scotland’s first minister had instigated a testy exchange with Farage on immigration. Sturgeon emerged from both unscathed. What’s more, she exited the broadcast looking like the only opposition leader able to articulate a credible anti-establishment position.

“I didn’t go into the debate intending to engage with Farage,” Sturgeon tells HuffPost, sitting in a meeting room on the top floor of Morgan Stanley’s headquarters overlooking Times Square in New York. Instead she resolved to draw her blade only if he was “offensive.” He was and she did, capping a bad night for the Ukip boss who was earlier booed for attacking the audience.

“I think Farage was exposed in those debates,” she reflects. “Once he’d blamed the foreigners, there was pretty much nothing else. I think people saw him for what he was.”

In the days following the debate, Sturgeon’s inbox was deluged by emails from English voters asking her to stand candidates south of the border, a flood she ascribes to a “deep disillusionment about the lack of choice on offer.”

Decrying the Tories, Labour and the Lib Dems as “different shades of the same thing,” the 44-year-old says the appetite for an alternative “is just as strong” in the rest of the UK as it is in Scotland. “But my goodness, there’s a market for a really social democratic party in England,” she suggests.

Yet a month later David Cameron returned to Downing Street, his tenure at Number 10 freshly unshackled from coalition. “They [the voters] opted for the devil they knew,” says Sturgeon. Her post election analysis is simple: Miliband failed to “do the deal with voters in England” and was therefore not a “viable alternative.”

The former solicitor quickly rebukes any notion that the SNP aided the Tories by splitting the opposition vote. “Labour could have won every seat in Scotland and they still wouldn’t have won the election because the failed to beat the Tories in England,” she says. “It’s not for me to answer why Labour failed in England, but that’s the question they’ve got to answer.”

Sturgeon admits the debates were “nerve-wracking” with “a lot riding on it for all of us.” Emblematic of the relief was the now-famed hug between Sturgeon and her fellow panellists, Natalie Bennett and Leanne Wood, as David Dimbleby closed the show.

Yet the SNP chief suggests it was more than just respite, calling it a “vivid illustration” of how women approach politics. “Three men wouldn’t have done that, even if they felt the same,” she says, recalling the positive reaction she received “to see women represented” and how it had “changed the tone.”

The day before our interview, Sturgeon was similarly anxious ahead of an appearance on “The Daily Show With Jon Stewart.” Despite being broadcast on Comedy Central, the show has become the sole proponent of oppositional political debate in the US, much to the shame of mainstream news outlets. As such it has a huge following nationally and around the world. Invitations to appear are not to be snubbed.

“I got through it,” she laughs, describing the show’s outgoing host as “charming and very well informed.” Her appearance was the centrepiece of a four-day tour of the US to promote business, tourism and study. Within the segment, the first minister was asked about the tribal nature of British politics and whether Blighty was becoming polarised, similar to the US.

She reflects further on this during our sit down, saying “it feels like it has” but attributes that to social media, that has given “people who want to be aggressive and abusive… ways of being heard.”

“The referendum, by its nature, had a ‘yes’ camp and a ‘no’ camp, which can make people feel as though they’re divided, but overwhelmingly the experience of the referendum on both sides was positive,” she says.

The SNP leader appears on Comedy Central’s ‘The Daily Show with Jon Stewart’, part of her four-day tour of the United States

Yet Sturgeon insists the invective doesn’t solely emanate from her corner of the Union, with people “on both sides” engaging in online abuse. “I could shown you the abuse that gets hurled at me on Twitter,” she says, nodding to her phone. Still, the first minister remains a fan of the micro-blogging site, which “democratises public debate” even if the tone is “impossible to control.”

“Party leaders do have a responsibility to speak out,” she says, “and I do that more than any other party leader. Even people who are professing to be on my side, I’ll call them out.”

The cybernats emerged during the bruising referendum campaign, culminating last September in a narrow defeat for the nationalists. Scotland was to remain in the Union and the groundswell of SNP support would quickly dissipate. Except, it didn’t. The momentum increased, propelling Sturgeon’s party to a landslide victory at the general election, securing 56 of the 59 Scottish seats.

“We are not deliberately going around Westminster trying to annoy people,” says Sturgeon. “We’ve got people in our group that are new to politics. They don’t know what the hundred-year traditions are. Clapping is commonplace in the Scottish parliament. Who knew it’s forbidden in the House of Commons?”

Still, she remains bullish, insisting the SNP is doing “what it should be doing — asserting its position as the third biggest party in the House of Commons.” Her MPs are not there to “be disruptive or destructive,” but are there “to get things done.”

Despite general election success, Sturgeon still regards the referendum as a “devastating defeat” though she notes it quickly became clear that “something had changed in Scotland” and the country would not go “back to the way we were before the referendum.” Having tasted “what it was like to be in charge of the destiny of our own country” that appetite would not be sated.

Yet victory in May created fresh problems for the party, whose ultimate ambition remains to leave the Union. How quickly can they call for a second vote on independence? Waiting has a generational benefit (young people are more likely to vote ‘yes’ so better to be patient) however delay risks diminishing the enthusiastic support that has propelled the nationalists to lofty heights. Sturgeon cuts a middle road, describing the “deep pragmatism” of the people who want independence.

“We know we can’t rush it,” she says. “We didn’t persuade the majority [at the referendum], and there’s no shortcut. You can’t just keep asking the question over and over again until you get the answer you want. You have to build a case through patient endeavour.” In the meantime, Sturgeon is determined to “get on with running the country,” including pushing for more devolution from Westminster, even more than was promised by the post-referendum Smith Report.

“Smith was a response by the Westminster parties to the referendum result,” she says. “They need to come up with a response to the general election result. To say it’s business as usual and carry on with ‘Smith’ won’t satisfy people.”

Sturgeon is unmoved by the argument that further devolution could undercut her quest for independence, machinations she dismisses as “Machiavellian.” She is more open to idea that an EU referendum could lead to a second independence vote, though admits her preference would be not to hold an EU vote at all. “I hope the UK votes to stay in,” she says. “If that doesn’t happen we’ll have to see. But it’s one scenario that could increase demand for a second referendum because Scotland is not going to look kindly on being taken out of the EU.”

We return to Labour and the question of whether the SNP needs a robust opposition party in Scotland if only for accountability. “I think it’s healthy in any democracy to have a strong opposition but as first minister I can’t create one,” she says. “Also, I’m still a politician. I’m not going to wish for the quick recovery of my main political opponent, not when I’ve got an election in 11 months time.”